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In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.
The regularized incomplete beta function is the cumulative distribution function of the beta distribution, and is related to the cumulative distribution function (;,) of a random variable X following a binomial distribution with probability of single success p and number of Bernoulli trials n:
where I is the regularized incomplete beta function. While the related beta distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed as a probability, the beta prime distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed in odds.
The Type I cumulative distribution function is usually represented as a Poisson mixture of central beta random variables: [1] = = (+,),where λ is the noncentrality parameter, P(.) is the Poisson(λ/2) probability mass function, \alpha=m/2 and \beta=n/2 are shape parameters, and (,) is the incomplete beta function.
Beta regression is a form of regression which is used when the response variable, , takes values within (,) and can be assumed to follow a beta distribution. [1] It is generalisable to variables which takes values in the arbitrary open interval ( a , b ) {\displaystyle (a,b)} through transformations. [ 1 ]
The beta-binomial is a one-dimensional version of the Dirichlet-multinomial distribution as the binomial and beta distributions are univariate versions of the multinomial and Dirichlet distributions respectively. The special case where α and β are integers is also known as the negative hypergeometric distribution.
For every odd positive integer +, the following equation holds: [3] (+) = ()!() +where is the n-th Euler Number.This yields: =,() =,() =,() =For the values of the Dirichlet beta function at even positive integers no elementary closed form is known, and no method has yet been found for determining the arithmetic nature of even beta values (similarly to the Riemann zeta function at odd integers ...
A Pearson density p is defined to be any valid solution to the differential equation (cf. Pearson 1895, p. 381) ′ () + + + + = ()with: =, = = +, =. According to Ord, [3] Pearson devised the underlying form of Equation (1) on the basis of, firstly, the formula for the derivative of the logarithm of the density function of the normal distribution (which gives a linear function) and, secondly ...
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